Algebra challenge worksheet for collecting like terms, organized by difficulty from 1 to 4 points.

Problem: Collect Like Terms

The task involves simplifying algebraic expressions by collecting like terms. Like terms are terms that have the same variable(s) raised to the same power. For example, \(3x\) and \(5x\) are like terms, but \(3x\) and \(5y\) are not.

Let's solve each expression step by step, starting from the 1-point section and moving up to the 4-point section.

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#### 1 Point (Simple Expressions)
1. \(a + a + 2a\)
- Combine all \(a\)-terms: \(1a + 1a + 2a = 4a\).
- Answer: \(\boxed{4a}\)

2. \(b + b\)
- Combine all \(b\)-terms: \(1b + 1b = 2b\).
- Answer: \(\boxed{2b}\)

3. \(2x + 3x\)
- Combine all \(x\)-terms: \(2x + 3x = 5x\).
- Answer: \(\boxed{5x}\)

4. \(7y - 5y + y\)
- Combine all \(y\)-terms: \(7y - 5y + 1y = 3y\).
- Answer: \(\boxed{3y}\)

5. \(5g - g + 2g\)
- Combine all \(g\)-terms: \(5g - 1g + 2g = 6g\).
- Answer: \(\boxed{6g}\)

6. \(5x - 3x - x\)
- Combine all \(x\)-terms: \(5x - 3x - 1x = 1x = x\).
- Answer: \(\boxed{x}\)

7. \(4d + 2d\)
- Combine all \(d\)-terms: \(4d + 2d = 6d\).
- Answer: \(\boxed{6d}\)

8. \(5k - k + 6k\)
- Combine all \(k\)-terms: \(5k - 1k + 6k = 10k\).
- Answer: \(\boxed{10k}\)

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#### 2 Points (Moderate Expressions)
1. \(4w - 5w + 2w\)
- Combine all \(w\)-terms: \(4w - 5w + 2w = 1w = w\).
- Answer: \(\boxed{w}\)

2. \(4x - 3x + y\)
- Combine \(x\)-terms: \(4x - 3x = 1x = x\).
- The \(y\)-term remains as it is.
- Answer: \(\boxed{x + y}\)

3. \(2x + 4y - 3x\)
- Combine \(x\)-terms: \(2x - 3x = -1x = -x\).
- The \(y\)-term remains as it is.
- Answer: \(\boxed{-x + 4y}\)

4. \(4t + 3s - t + 2s\)
- Combine \(t\)-terms: \(4t - 1t = 3t\).
- Combine \(s\)-terms: \(3s + 2s = 5s\).
- Answer: \(\boxed{3t + 5s}\)

5. \(3g + h - 2g\)
- Combine \(g\)-terms: \(3g - 2g = 1g = g\).
- The \(h\)-term remains as it is.
- Answer: \(\boxed{g + h}\)

6. \(5x - 2y + x\)
- Combine \(x\)-terms: \(5x + 1x = 6x\).
- The \(y\)-term remains as it is.
- Answer: \(\boxed{6x - 2y}\)

7. \(3c - 2d + c\)
- Combine \(c\)-terms: \(3c + 1c = 4c\).
- The \(d\)-term remains as it is.
- Answer: \(\boxed{4c - 2d}\)

8. \(3e + 2f + e + 4f\)
- Combine \(e\)-terms: \(3e + 1e = 4e\).
- Combine \(f\)-terms: \(2f + 4f = 6f\).
- Answer: \(\boxed{4e + 6f}\)

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#### 3 Points (Complex Expressions)
1. \(3c - 4d + c + d\)
- Combine \(c\)-terms: \(3c + 1c = 4c\).
- Combine \(d\)-terms: \(-4d + 1d = -3d\).
- Answer: \(\boxed{4c - 3d}\)

2. \(5g + 2h - 4g + h\)
- Combine \(g\)-terms: \(5g - 4g = 1g = g\).
- Combine \(h\)-terms: \(2h + 1h = 3h\).
- Answer: \(\boxed{g + 3h}\)

3. \(2x + 3y - 4x - y\)
- Combine \(x\)-terms: \(2x - 4x = -2x\).
- Combine \(y\)-terms: \(3y - 1y = 2y\).
- Answer: \(\boxed{-2x + 2y}\)

4. \(4g + h - 3g - 5h\)
- Combine \(g\)-terms: \(4g - 3g = 1g = g\).
- Combine \(h\)-terms: \(1h - 5h = -4h\).
- Answer: \(\boxed{g - 4h}\)

5. \(3g + h - 2g\)
- Combine \(g\)-terms: \(3g - 2g = 1g = g\).
- The \(h\)-term remains as it is.
- Answer: \(\boxed{g + h}\)

6. \(6j - 4k + j - k\)
- Combine \(j\)-terms: \(6j + 1j = 7j\).
- Combine \(k\)-terms: \(-4k - 1k = -5k\).
- Answer: \(\boxed{7j - 5k}\)

7. \(9c - 6j - 10h + j\)
- Combine \(c\)-terms: \(9c\) (no other \(c\)-terms).
- Combine \(j\)-terms: \(-6j + 1j = -5j\).
- Combine \(h\)-terms: \(-10h\) (no other \(h\)-terms).
- Answer: \(\boxed{9c - 5j - 10h}\)

8. \(4u + v - 5u + v\)
- Combine \(u\)-terms: \(4u - 5u = -1u = -u\).
- Combine \(v\)-terms: \(1v + 1v = 2v\).
- Answer: \(\boxed{-u + 2v}\)

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#### 4 Points (Challenging Expressions)
1. \(3.5x - y + 2.5x\)
- Combine \(x\)-terms: \(3.5x + 2.5x = 6x\).
- The \(y\)-term remains as it is.
- Answer: \(\boxed{6x - y}\)

2. \(1.5g - 2.5h + h\)
- The \(g\)-term remains as it is.
- Combine \(h\)-terms: \(-2.5h + 1h = -1.5h\).
- Answer: \(\boxed{1.5g - 1.5h}\)

3. \(2x - 3.5y - 3.5x\)
- Combine \(x\)-terms: \(2x - 3.5x = -1.5x\).
- The \(y\)-term remains as it is.
- Answer: \(\boxed{-1.5x - 3.5y}\)

4. \(2t - 6s - 4t + s\)
- Combine \(t\)-terms: \(2t - 4t = -2t\).
- Combine \(s\)-terms: \(-6s + 1s = -5s\).
- Answer: \(\boxed{-2t - 5s}\)

5. \(3e - 1.25f + f\)
- The \(e\)-term remains as it is.
- Combine \(f\)-terms: \(-1.25f + 1f = -0.25f\).
- Answer: \(\boxed{3e - 0.25f}\)

6. \(-4x + 2.5y + 2.5x\)
- Combine \(x\)-terms: \(-4x + 2.5x = -1.5x\).
- The \(y\)-term remains as it is.
- Answer: \(\boxed{-1.5x + 2.5y}\)

7. \(5s - 2.25t + 1.5t\)
- The \(s\)-term remains as it is.
- Combine \(t\)-terms: \(-2.25t + 1.5t = -0.75t\).
- Answer: \(\boxed{5s - 0.75t}\)

8. \(-2g + 2.5h + 1.5g\)
- Combine \(g\)-terms: \(-2g + 1.5g = -0.5g\).
- The \(h\)-term remains as it is.
- Answer: \(\boxed{-0.5g + 2.5h}\)

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Final Answers:

- 1 Point: \(\boxed{4a, 2b, 5x, 3y, 6g, x, 6d, 10k}\)
- 2 Points: \(\boxed{w, x + y, -x + 4y, 3t + 5s, g + h, 6x - 2y, 4c - 2d, 4e + 6f}\)
- 3 Points: \(\boxed{4c - 3d, g + 3h, -2x + 2y, g - 4h, g + h, 7j - 5k, 9c - 5j - 10h, -u + 2v}\)
- 4 Points: \(\boxed{6x - y, 1.5g - 1.5h, -1.5x - 3.5y, -2t - 5s, 3e - 0.25f, -1.5x + 2.5y, 5s - 0.75t, -0.5g + 2.5h}\)

Final Boxed Answer:
\[
\boxed{
\begin{array}{l}
\text{1 Point: } 4a, 2b, 5x, 3y, 6g, x, 6d, 10k \\
\text{2 Points: } w, x + y, -x + 4y, 3t + 5s, g + h, 6x - 2y, 4c - 2d, 4e + 6f \\
\text{3 Points: } 4c - 3d, g + 3h, -2x + 2y, g - 4h, g + h, 7j - 5k, 9c - 5j - 10h, -u + 2v \\
\text{4 Points: } 6x - y, 1.5g - 1.5h, -1.5x - 3.5y, -2t - 5s, 3e - 0.25f, -1.5x + 2.5y, 5s - 0.75t, -0.5g + 2.5h
\end{array}
}
\]